the triangle below is equilateral. find the length of side $x$ in simplest radical form with a rational…

Explanation:

Step1: Identify right triangle sides

The right triangle has hypotenuse $\sqrt{10}$ and one leg $x$. In an equilateral triangle, the altitude splits the side into two equal parts, so this right triangle is a 30-60-90 triangle where $x$ is the shorter leg (adjacent to the 60° angle).

Step2: Use cosine of 60°

$\cos(60^\circ) = \frac{x}{\sqrt{10}}$, and $\cos(60^\circ) = \frac{1}{2}$

Step3: Solve for x

Rearrange the equation: $x = \sqrt{10} \times \cos(60^\circ) = \sqrt{10} \times \frac{1}{2}$ Simplify to rational form: $x = \frac{\sqrt{10}}{2}$

Answer:

$\frac{\sqrt{10}}{2}$